{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-statistical-estimator-01",
  "mko_id": "mko-statistical-estimator",
  "title": {
    "zh-Hant": "同一資料函數與兩個參數law",
    "en": "One data function under two parameter laws"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-statistical-estimator-01",
    "source": {
      "mko_id": "mko-statistical-estimator",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "ba7fbdff9e10ebc9977253d4edccfcba77e5e47c0404aae6d00ad578d8bf6d2f",
      "english_sha256": "d391d517379756c16fdbbb5518416c50d6391c8b6b189ea66e1178dbe90b2c2a"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "核total estimator並計算逐theta期望。",
      "en": "Validate a total estimator and its parameterwise expectations."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Θ={1/4,3/4}，Ω={00,01,10,11}；P_(1/4)=(9,3,3,1)/16，P_(3/4)=(1,3,3,9)/16。",
          "en": "Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "U(x)=(x1+x2)/4，依Ω順序的total table為(0,1/4,1/4,1/2)，目標參數為g(theta)=theta。",
          "en": "U(x)=(x1+x2)/4 has total table(0,1/4,1/4,1/2) in Omega order, with target g(theta)=theta."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2",
          "work"
        ],
        "before_tex": "\\Omega=\\{00,01,10,11\\}",
        "after_tex": "U=(0,1/4,1/4,1/2)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個outcome恰有一值，完整table給total函數。",
          "en": "Every outcome has exactly one value, giving a total function."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有漏掉11或多值entry。",
            "en": "Neither11 nor any single-valuedness check is omitted."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01",
          "work"
        ],
        "before_tex": "U(x)=(x_1+x_2)/4",
        "after_tex": "U\\text{ does not depend on }\\theta",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "規則只讀兩個觀察bit，不讀未知參數。",
          "en": "The rule reads only the observed bits, not the unknown parameter."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個theta都使用同一個U table。",
            "en": "Both theta values use the same U table."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "G2",
          "S02",
          "work"
        ],
        "before_tex": "x=01",
        "after_tex": "u_{\\rm obs}=U(01)=1/4",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "estimate是函數在此次資料的值。",
          "en": "The estimate is the function's value at this observation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "estimate1/4不是整個四值函數U。",
            "en": "The value 1/4 is not the entire four-entry function U."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "G2",
          "S03",
          "work"
        ],
        "before_tex": "E_{1/4}[U]=0(9/16)+(1/4)(3/16)+(1/4)(3/16)+(1/2)(1/16)",
        "after_tex": "E_{1/4}[U]=1/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在第一law完整加權，兩個中間outcomes都保留。",
          "en": "Weight all outcomes under the first law, retaining both middle outcomes."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "01與10值相同但各自有質量。",
            "en": "01 and 10 share a value but each contributes mass."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "S04",
          "work"
        ],
        "before_tex": "E_{3/4}[U]=0(1/16)+(1/4)(3/16)+(1/4)(3/16)+(1/2)(9/16)",
        "after_tex": "E_{3/4}[U]=3/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二law用9/16給11，而非沿用第一列。",
          "en": "The second law assigns9/16 to 11, rather than reusing the first row."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不把不同theta的權重混合。",
            "en": "Do not mix weights from different theta laws."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "G2",
          "S05",
          "work"
        ],
        "before_tex": "E_{1/4}[U]=1/8,\\quad E_{3/4}[U]=3/8",
        "after_tex": "\\theta\\mapsto E_\\theta[U]\\text{ is a parameterwise table}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同函數的分布性質可隨theta改變。",
          "en": "Distributional properties of the same function may vary with theta."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "變的是law而不是estimator定義。",
            "en": "The law changes, not the estimator definition."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2",
          "S06",
          "work"
        ],
        "before_tex": "V_\\theta(x)=\\theta",
        "after_tex": "V_\\theta\\text{ is not a usable unknown-parameter estimator rule}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若為不同未知theta換常數規則，不能由資料執行。",
          "en": "Changing the constant rule with unknown theta does not define a data-executable estimator."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "形式上每個Vtheta是函數，但不是此模型要求的同一資料函數。",
            "en": "Each fixed Vtheta is a function, but not the one common data function required here."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "G2",
          "S07",
          "work"
        ],
        "before_tex": "U:\\Omega\\to\\mathbb Q\\text{ total and parameter-independent}",
        "after_tex": "U\\text{ qualifies as an estimator for this model}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完成函數資格；bias、variance、sufficiency須各另核。",
          "en": "The functional qualification is complete; bias, variance and sufficiency need separate audits."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個期望數字沒有被擴大成一致性或posterior結論。",
            "en": "The two expectation values are not inflated into consistency or posterior claims."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "U是同一total estimator；觀察01的estimate為1/4，兩law期望為1/8與3/8。",
      "en": "U is one total estimator; at 01 the estimate is 1/4, while its two law expectations are 1/8 and 3/8."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "U在11沒定義也可稱total。",
          "en": "U may be total without a value at 11."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "total要求每個carrier outcome有值。",
          "en": "Totality requires a value for every carrier outcome."
        }
      },
      {
        "claim": {
          "zh-Hant": "令U(x)=theta即可做未知theta的資料估計。",
          "en": "Set U(x)=theta to estimate unknown theta from data."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "未知theta不能作資料規則的輸入。",
          "en": "The unknown parameter cannot be an input to the data rule."
        }
      },
      {
        "claim": {
          "zh-Hant": "觀察estimate1/4就是E_(1/4)U。",
          "en": "The observed 1/4 equals E_(1/4)U."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "完整加權期望為1/8。",
          "en": "The complete expectation is 1/8."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "有限parameter model與共同sample space上，statistic table只依observation。逐theta可計算expectation但本輪不宣稱unbiased、consistent、sufficient、identifiable或posterior；完整sampling distribution留後續節點。",
          "en": "On a finite parameter model with one common sample space, the statistic table depends only on the observation. Expectations can be computed under each parameter law, but no unbiasedness, consistency, sufficiency, identifiability or posterior claim follows here. Full sampling distributions are a later object."
        },
        "source": {
          "mko_id": "mko-statistical-estimator",
          "sha256": "ba7fbdff9e10ebc9977253d4edccfcba77e5e47c0404aae6d00ad578d8bf6d2f",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限law機率非負且總和一；statistic只依樣本，期望為所有outcomes的加權和。",
          "en": "Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(1/4)*(6/16)+(1/2)*(1/16)",
        "rhs": "1/8"
      },
      {
        "id": "N02",
        "lhs": "(1/4)*(6/16)+(1/2)*(9/16)",
        "rhs": "3/8"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "本節不由期望自動宣稱統計品質。",
      "en": "Expectations here do not automatically establish estimator quality."
    },
    "md_symbols": [
      [
        "θ",
        "固定但未知參數",
        "fixed unknown parameter"
      ],
      [
        "U",
        "同一total statistic",
        "one total statistic"
      ]
    ],
    "md_error_controls": [
      {
        "title": "total table / Total table",
        "zh": "四個outcomes逐一對應，不因同值合併而漏資料。",
        "en": "Assign every outcome even when values coincide."
      },
      {
        "title": "theta與資料 / Parameter and data",
        "zh": "同一U，兩套Ptheta，禁止暗換U。",
        "en": "Keep one U and two Ptheta laws; do not silently change U."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "7d2a71efe1386b0a4e2ae04a859c98e82bdbfb5e2aa1303f1a80575333c06239",
    "original_source_sha256": "17561f6cd64c683c14d172706c4eb1d4846352d8213e1cf1b580c63cbb581dce",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S072",
      "BR142-S073",
      "BR142-S074",
      "BR142-S071",
      "BR142-S068",
      "BR142-S069",
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    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}